Spherical fuzzy PROMETHEE (Sharaf, 2021)

Four things change; the logic of entering and leaving flow does not.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell is three numbers: a degree of support (μ), a degree of rejection (ν) and a degree of hesitancy (π), the sum of whose squares cannot exceed 1. Hesitancy is not a residual left over, as it is in intuitionistic or Pythagorean structures; the expert supplies it separately.

An unusual intermediate step: relative closeness. Direct subtraction is not defined between spherical fuzzy numbers. So a closeness ratio is first computed for every cell against an ideal and an anti-ideal point, as in crisp TOPSIS. The difference between two alternatives is the crisp difference between these two closeness ratios; the spherical fuzzy numbers themselves are never subtracted.

A six-rung preference function. In crisp PROMETHEE the preference function is usually defined through thresholds (indifference, preference). Here the closeness difference is mapped, by its magnitude, onto one of six verbal rungs (very weak, weak, moderate, strong, very strong, absolute preference), and each rung has its own spherical fuzzy counterpart. If the difference is negative, the complementary rung is used.

Flows stay spherical fuzzy; the score is taken only at the very end. The leaving and entering flows are sums of these spherical fuzzy preference cells and are themselves spherical fuzzy numbers. The net flow is only crisped at the very last step, by scoring these two flows.

DecisionMind holds the base method's leaving/entering-flow summation logic fixed in this extension; what changes is how a preference turns into a cell and how those cells are summed.

What stays the same as the base method is that the result is again a net flow and a rank. The difference is that this net flow comes from the score of spherical fuzzy flows that have already passed through a six-rung preference scale.

When the difference is large (a closeness gap above one half), the preference rung settles at the extreme, absolute preference. If several pairs of alternatives show a large gap, the flows can end up very close to one another; a small gap between net flows does not then mean the alternatives are genuinely close, it can mean the scale has saturated at its extreme.

Thus instead of writing:

"The net flows are very close, so the alternatives are of equal value"

the report should read:

"Because the closeness gaps are large, the preference rung has settled at the extreme in most pairs; the net flows coming out close together shows that the scale has saturated, not that the alternatives are equivalent"

Use this when the expert's hesitancy is information asked for and reported separately, rather than whatever is left over after support and rejection. The spherical fuzzy data-type card sets out this distinction in detail; if hesitancy has not been measured, this extension is not needed.

If the three degrees already fit within the picture fuzzy constraint (summing to at most 1), there is no need to move to the spherical structure; the relevant PROMETHEE extension is enough. The spherical structure should not be confused with the Pythagorean or q-rung structures: in the spherical structure a third degree, hesitancy, comes directly from the expert, whereas in the others it does not.

Entering data without checking the value-space constraint. This is the manifest's own warning: every cell must satisfy μ² + ν² + π² ≤ 1, and this must be verified before the calculation runs.

Interpreting the preference rung as saturated without noticing. If closeness gaps are large, most pairs fall into the absolute-preference rung and the flows converge; this should be read as "the scale has saturated," not as "the alternatives are equivalent."

Changing the score function without noticing. This is the manifest's own warning: more than one score function exists for reducing a spherical fuzzy number to a single number, and the one chosen affects the result; which function was used must be stated in the report.

Crisping spherical fuzzy flows too early. If the leaving and entering flows are reduced to a crisp number before the net flow is calculated, the three-degree information carried in the intermediate step is lost; crisping happens only at the very last step.

The governing principle is this:

Spherical fuzzy PROMETHEE is for situations where the three degrees come from the expert; but because subtraction is not defined between spherical numbers, preference is mapped onto a closeness gap, and that in turn onto a six-rung scale. This scale's saturation point must be taken into account when reading the result.